The present invention relates generally to data processing systems. More particularly, the present invention relates to data processing systems for performing logical operations based on fuzzy information.
Throughout history, mathematics evolved through increasingly rich systems of numbers, beginning with the positive integers. Later extensions added the concept of zero, then negative integers, ratios of integers (the rationals), the irrational numbers, the addition of infinity, and finally the complex number system.
In 1865, for example, Boltzmann provided the first full explanation and mathematical analysis of the available energy in a closed physical system with the definition of entropy, a measure of the disorder of a physical system. For the last 140 years the disciplines of statistical mechanics and thermodynamics have advanced these concepts in the areas of physics, chemistry, and engineering.
In 1948, Shannon wrote the fundamental paper on the science of information theory with definitions of the information content (“negative entropy”) of a string of symbols, and additional definitions of the concepts of information redundancy and transmission loss. These foundational concepts on entropy and information were founded in part upon the previous work of Boltzmann and specifically upon the fact that highly organized energy in a physical system, and highly organized structures of symbols in an information system, are very improbable states. As time evolves, if a system is allowed to visit all physical or information states equally, then the systems will spend time in states in proportion to the probability of that state existing. Thus energy that is concentrated in a small region or in just a few particles becomes evenly dispersed over the whole system and unavailable for useful purposes. Likewise the order of strings of symbols becomes randomized and thus contains less and less information. In order for the entropy (or information) of two systems to be additive, then in view of the fact that the probabilities of given states are multiplicative, they were led to the definition of both entropy and information in terms of the logarithm of the probability of a states occupation.
In 1905, Markov developed a theory of discrete matrix transformations that transform a vector in a way that preserves the sum of the components (as opposed to the sum of the squares or other invariant form). This is valuable because it is applicable to a vector of probabilities (such as the probabilities that an object might be in any one of 12 locations) and thus where the sum of these values must give a total of 1 (certainty) to be somewhere. Likewise the Markov theory is applicable to the study of a fixed total number of entities where a Markov transformation shifts the number of objects in each category but leaves the total number of objects invariant.
In 1985, the present inventor extended the foundation of Markov's work to a theory of continuous transformations (a Markov-Type Lie Group and generating Algebra) by showing that the set of all continuous linear transformations (the General Linear Group in n dimensions) can be decomposed into Markov-type transformations and another set of continuous transformations that simply multiply each variable by a scale factor thus providing growth or diminution of the associated values. A particular transformation was found to give diffusion of a system and thus to increase entropy (and equivalently reduce information content) thereby relating increasing entropy to evolution under a group of transformations.
In approximately 1202, Fibonacci studied the prediction of populations of rabbits using a mathematical model in which he proposed that each pair of rabbits would reproduce a new pair every unit of time (except the first unit of time while they are maturing) and with the assumption that no rabbits die. Beginning with one pair of rabbits at t=1, then one has a 1 pair again at t=2, then 2 pair (the older pair and their offspring). The sequence is: 1,1,2,3,5,8,13, where n(t)=n(t−1)+n(t−2). This sequence has come to be known as the Fibonacci sequence. During the 18th and 19th centuries, the Fibonacci number sequence was observed to occur in nature in pinecones, sunflowers and other living systems leading to unresolved conjectures as to why this sequence appears in living things.
At the same time, it was noticed that the ratio of two adjacent Fibonacci numbers approached a limit, 1.618 . . . , called the Golden Mean that appeared in architecture, the human form, and frequently in art. It was even suggested that the Golden Mean was a number third in importance only to e and to Pi. The present inventor was able to show that the Fibonacci sequence could be written as a combination of elements in the Markov Lie Group and certain scaling transformations in a particular ratio. He found a function that interpolated the Fibonacci sequence (much like the gamma function interpolates the discrete values of the factorial). Associated differential equations for this function were found and it was shown that the Fibonacci sequence, associated continuous function, differential equations, and Lie Group generating transformations were all a part of an infinite sequence of similar types of numerical sequences and functions of which the Fibonacci sequence was the “simplest.”
Specifically, the Lie Algebra generating the Fibonacci sequence was found to be a specific combination of diffusion (entropy increasing/information decreasing) transformations and growth (scaling of the coordinates). An argument was presented that the Fibonacci sequence was the simplest possible transformation, on a two state system, for which diffusion of information was exactly countered by growth in such a way that allowed a simple system to retain information by growth in order to counter increasing entropy. But a proper mathematical framework for the definition of information in such a system was not available at the time.
Principia Mathematica was written in three volumes from 1910 to 1913 by Russell and Whitehead laying out the foundations of mathematics upon a rigorous foundation of logic. In particular, propositional calculus lays out a theory of variables and rules for combining these variables that can take on the binary values of true or false with operations including “negation,” “and,” “or,” “implies,” and “equivalence.” Such a framework is needed in order to formalize the meaning of language and statements in the areas known as Predicate Calculus and Syllogistic Logic and Set Theory thereby laying the foundations for both Mathematics and Logic that began with the work of Boole in the 1820s. The elements in these theories are variables that are combined using tables to determine the result of the combination of two values by any of the applicable binary operations (or the unary operation of negation acting on one value). These variables only have the values of T and F (or 1 and 0).
The last 50 years has seen a complete paradigm shift under the developments issuing from VonNeuman's invention in 1945 of the stored program computer and the ensuing evolution of machines that rapidly manipulate symbols along with the symbolic structures and the computer software languages and structures to support these advances. Foundational to these advances are the combinations of the symbols “1” and “0” (termed a bit of information). Of special interest are the new advances in quantum computing where one might think of particle spin (up or down such as an electron) as representing the bit values “1” and “0”. In spite of the fact that all scientific measurements have limited accuracy and inherent numerical uncertainty, computers still execute sequential decision paths with exact numerical values.
“Information” is currently understood at the “intermediate level” as unique strings of characters, specifically, with character “bytes” represented as eight binary 1/0 bits and thus information is expressed in the most basic form as ordered strings of 1's and 0's. The number of bits (1/0) provides the measure of information content as based upon the Shannon theory. However, the “higher levels” of information theory, such as how to measure the information content of a sentence, equation, industrial process, work of art (book, art, and music), has proven to be relatively intractable.
The common thread in the works just listed is that they all address the measurement, representation, and management of information either under the conditions of exact knowledge or under conditions of numerical uncertainty. Work over the last 50 years in “fuzzy logic” has not pursued the paths outlined in this application because the condition of mathematical distributivity among certain logical operations is not valid and thus alternative directions have been pursued by mathematicians in the area known as “fuzzy mathematics”. U.S. Pat. No. 5,924,085, incorporated herein by reference, shows a device operating according to “fuzzy logic” principles.